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03 Sept. 2007
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Fun With Mirrors

You've heard about carbon dioxide by now.  It's ubiquitous.  Not only does your car create it when burning gasoline, you create it when burning food.  Not to mention non-nuclear electric plants and a vast number of other human activities.  Carbon dioxide is a "greenhouse gas," and many scientists believe that this "anthropogenic" CO2 we are emitting will usher in a new era of "global warming" in which the ice caps will melt and vast, mostly deleterious, climate changes will result.  It will be our fault, dammit; we'd better do something about it now.  (Other scientists aren't so sure.  Freeman Dyson, for one, is skeptical of the climate scientists' "models," so if you're skeptical, too, you're in good company.)

If you like to drive, or, for that matter, breathe, you may find it difficult to reduce your CO2 output.  I cut mine in half, at least as far as driving is concerned, by buying a Prius hybrid.  But still it's only in half.  What if I'm a serious believer in the threat, and want to reduce it to zero?  No problem!  There are organizations that sell "offsets."  While most human activities create more carbon dioxide, some reduce it.  Planting trees, increasing energy efficiency of some other process, "sequestering" the gas underground, all reduce its atmospheric burden.  These organizations accomplish it in various ways.  You accomplish it by surprise! sending them money.

Why Did the Chicken Cross the Road?

The road, in this case, is the kind of road you are likely to find in front of your house.  It's two lanes, separated by a yellow or double-yellow line down the center and two white "fog lines" demarking its edges.  It's paved with asphalt, often called "black top" because let's not always see the same hands it's black.  The chicken crossed because once he stepped on it his feet got very hot, and, even with a brief respite gained by pausing at the yellow lines, he wanted to get to the other side and cool them off.  Clever chicken!  I made the same discovery one day when the shoe police were unaccountably preoccupied and thought to myself "If they made roads like this in 'inverse video' there would be a lot less heat absorbed by the planet."  But forget that, because this is a blogitem about mirrors.

Sending them Money

A quick Google search located a carbon footprint calculator, which in turn yielded the amount of money I would have to spend to compensate for the CO2 spewing forth from my Prius every year.  It was about $45.  I would have sent them a check right away, but that would mean they'd have 45 of my dollars and I wouldn't.  Instead I thought of an experiment.  The global warming problem is (allegedly) occurring because CO2, a major culprit in the greenhouse effect, is causing solar radiation that would otherwise escape back into space to instead warm the planet.  How about if I were to keep the $45, and instead put a mirror someplace where it would send that nasty radiation back to the sun?  Although I don't have a mirror, and may not have any good place to put it without complaints about the glare, I can at least think about it, right?  I'm curious to find out just how big a mirror it would require to accomplish this, and I can tell you honestly that, at the moment this is being written I have no idea.  But by the end of this blogitem you and I shall know if this makes any sense at all.

Warning: Calculations Ahead

How much CO2 does my Prius emit on an annual basis?  Easy!  I know I used 425.2 gallons of gas during calendar 2006, thanks to my trusty spreadsheet.  At 19.4 pounds per gallon, that's 4.12 tons of carbon dioxide.  How much energy have I gotten from this gas?  Each gallon contains 121MJ (Megajoule or million watt-seconds) which is the equivalent of 121,000,000/(3600*1000) = 33.6 kilowatt hours.  (Although I haven't "gotten" all this energy to move my car, one way or the other it has been added to the environment.)  Thus, to make my car "energy neutral," I have to somehow compensate for those tons of carbon dioxide and, although it is rarely mentioned as a problem, the energy I have added to the environment as well.  And I have to do it, me being me, without sending someone else money. 

I was about to perform a long calculation here about how much energy the sun deposits on the earth, but I was saved the trouble.  Quoting from the source,

Solar power per m2 on U.S. surface …this seems a little low…it’s 1342 watts per m2 outside the atmosphere, about 1000 watts per m2 at high noon on the ground, and on average (day and night) about 240 watts per meter2 absorbed at the ground. This is the average over the Earth too. 

I'm going to assume that "absorbed at the ground" implies an albedo (reflectance) of zero.  I'm also going to assume that this is correct for my location:  At the equator it would be higher, at the poles it would be lower.  At mid-latitude, it's probably close enough.  If I want to get rid of the energy I have added to the earth's environment, all I have to do is radiate 452.2*33.6kWh of energy back into space.  This is 15,194kWh and there are 8,760 hours in a year.  If I can dump 240 watt-hours per hour per square metre, I could put a mirror over a black spot on the ground and if all went well (which of course it won't), I would only need a 7.23 square metre mirror to compensate for my energy consumption!  Unfortunately this is over-optimistic.  For one thing, unless I kept pointing the mirror at the sun, I would only get the full benefit of its size when the sun is "overhead."  (I would at least try to tilt the mirror to a position appropriate for my latitude.)  So I'm going to add an efficiency multiplier of .7 for this effect.  Next, although the average 240W figure accounts for the energy hitting the mirror, it doesn't help with the energy being re-radiated to space.  This is close to 100% if the sky is clear, but much less if the energy is absorbed in the clouds.  With the justification that half of the reduction from 1000 to 240 (above) is caused by night, I'll assume the other half is caused by clouds and estimate an additional efficiency multiplier of .5.  This is a very rough figure and makes irrelevant all the decimal places in the other calculations.  Even so, assuming it is approximately correct, we now have our answer:

7.23 * (1/.7) * (1/.5) = 20.6.  Accordingly, a mirror of 20.6 square metres, or about 14.7 feet on a side would be sufficient.

Mirrors and Albedo

After estimating the mirror size (which I promise I didn't know before starting the calculation!), I realize that it's not entirely practical.  Not that I can't find a 14.7 foot square spot to put it, but rather that I would have to send money to somebody else to get the mirror. 

Sample albedos
Surface Typical
Albedo
Fresh asphalt 0.04[1]
Conifer forest
(Summer)
0.08[2]
Worn asphalt 0.12[1]
Bare soil 0.17[3]
Green grass 0.25[3]
Desert sand 0.40[4]
New concrete 0.55[3]
Fresh snow 0.80–0.90[3]

But all is not lost!  From the Wikipedia article on albedo comes the table at left.  A clean mirror will have an albedo as close to 1 as matters.  "Worn asphalt" as is present on my driveway and many others, has an albedo of .12.  It seems entirely possible to achieve the albedo of "fresh snow" artificially.  Striping paint should do the job, and on a driveway it will get little wear and so stay clean and reflective.  Re-estimating how much area would need to be raised from an albedo of .12 to .8 instead of the hypothetical effectiveness of the mirror (from 0 to 1), I come up with about 30 square metres instead of 20.

I'll still have to send money to somebody for the striping paint, but I bet it will be a lot less than the cost of the mirror.  I'm going outside now to measure my driveway.  I know just from thinking about it that it's far more than 30 square metres, but if you've been reading this attentively, you will realize that I "forgot" something important.  I still haven't "offset" the carbon dioxide!  This is getting a bit long, and I think the "Fun With Mirrors" title should be changed to "Fun with Striping Paint" but I won't do that.  Instead, I'll take care of the carbon dioxide in Part II, appearing here soon.  And we will have some fun with mirrors.  I promise.


NP:  "Black Mirror" - The Arcade Fire  (Coincidence!)

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Richard Factor

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